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      Questions tagged [cv.complex-variables]

      Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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      1answer
      46 views

      Riemann $P$-symbol for ODEs

      Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)? Examples: $P\...
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      1answer
      123 views

      Books on complex analysis for self learning that includes the Riemann zeta function?

      I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following: Analytic number theory : the connection between complex analysis and ...
      10
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      1answer
      329 views

      turn $\pi/n$, move $1/n$ forward

      start at the origin, first step number is 1. turn $\pi/n$ move $1/n$ units forward Angles are cumulative, so this procedure is equivalent (finitely) to $$ u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}...
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      1answer
      304 views

      Collinear Galois conjugates

      This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me. Let $p$ be an irreducible polynomial with integer ...
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      1answer
      53 views

      On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question

      This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$ According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...
      2
      votes
      2answers
      140 views

      On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

      Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above. Is $I_s$ known to be convergent for any real number $s<1$ ?
      6
      votes
      0answers
      282 views

      Cheap bound on $\zeta'(s)/\zeta(s)$ or $L'(s,\chi)/L(s,\chi)$?

      Say you are proving an explicit formula for $L(s,\chi)$ and/or the prime number theorem (in arithmetic progressions or not) in the usual way -- that is, shifting a line of integration from $\Re(s) = 1^...
      3
      votes
      1answer
      71 views

      Singular set of entire functions

      Recently I've started studying the theory of singular values for entire functions so I'm far from being a specialist in this field. In the literature I came across the following results: In [1] Gross ...
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      133 views

      Holomorphic functions to complex torus

      Let $X$ be a complex algebraic variety and $T$ a complex torus (not necessarily algebraic). Assume that $X$ is a proper subset of its completion $\bar{X}$. Let $f:X \to T$ be a holomorphic map. Are ...
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      0answers
      31 views

      A relaxed form of quantitative transversality

      Let $f$ be a holomorphic function on the unit ball $B_1(0) \subseteq \mathbb{C}^n$. Then given any pair of constants $\delta, \epsilon > 0$, does there exist a smooth function $g: B_1(0) \to \...
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      227 views

      How strange are Hölder domains?

      Let $D$ be a Jordan domain. We assume that $D$ is a H?lder domain. Namely, there exists a H?lder continuous Riemann map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk. It ...
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      0answers
      94 views

      How to calculate the contour integration?

      I encountered some problems when I read the paper. The details are shown as follows. \begin{equation} pdf(I)=e^I \prod_{i=1}^n \left(-j a_i\right)\int \frac{dk}{2\pi}\frac{e^{jk\left(e^I-1\right)}}{\...
      13
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      1answer
      295 views

      Does the $\overline{\partial}$ operator have closed image?

      Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
      1
      vote
      1answer
      60 views

      Disk with punctures and convex geodesical hull of the punctures isomorphic?

      Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary. Let us call this surface $X$. As it is well known, the disk can be equipped with an hyperbolic metric and is then ...
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      42 views

      Looking for a simple proof of approximating almost periodic functions by generalised trigonometric polynomials with restricted Bohr spectrum

      Let $AP$ be the space of almost periodic functions, defined to be the uniform closure in $C_b(\mathbb R,\mathbb C)$ of the set of generalized trigonometric polynomials $Q(t):=\sum_{j=1}^n a_j e^{i\...

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